L10n31

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(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n31 at Knotilus!

Link Presentations

[edit Notes on L10n31's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X4,19,1,20 X11,14,12,15 X3,10,4,11 X5,13,6,12 X13,5,14,20 X16,9,17,10 X15,2,16,3 X8,17,9,18
Gauss code	{1, 9, -5, -3}, {-6, -1, 2, -10, 8, 5, -4, 6, -7, 4, -9, -8, 10, -2, 3, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(v-2)(2v-1)(uv+1)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	${\frac {7}{q^{9/2}}}-{\frac {6}{q^{7/2}}}+{\frac {5}{q^{5/2}}}-{\frac {5}{q^{3/2}}}-{\frac {2}{q^{15/2}}}+{\frac {3}{q^{13/2}}}-{\frac {5}{q^{11/2}}}-{\sqrt {q}}+{\frac {2}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}z^{-1}-z^{3}a^{7}-3za^{7}-2a^{7}z^{-1}+z^{5}a^{5}+3z^{3}a^{5}+3za^{5}+a^{5}z^{-1}+z^{5}a^{3}+2z^{3}a^{3}+a^{3}z^{-1}-z^{3}a-2za-az^{-1}$ (db)
Kauffman polynomial	$3a^{9}z^{3}-5a^{9}z+a^{9}z^{-1}+a^{8}z^{6}+a^{8}+2a^{7}z^{7}-6a^{7}z^{5}+15a^{7}z^{3}-12a^{7}z+2a^{7}z^{-1}+a^{6}z^{8}+3a^{6}+4a^{5}z^{7}-9a^{5}z^{5}+11a^{5}z^{3}-8a^{5}z+a^{5}z^{-1}+a^{4}z^{8}+a^{4}z^{6}-4a^{4}z^{4}+2a^{4}+2a^{3}z^{7}-2a^{3}z^{5}-4a^{3}z^{3}+2a^{3}z-a^{3}z^{-1}+2a^{2}z^{6}-4a^{2}z^{4}+a^{2}+az^{5}-3az^{3}+3az-az^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 χ 2                 1 1 0               1   -1 -2             4 1   3 -4           2 2     0 -6         4 3       1 -8       3 2         -1 -10     2 4           -2 -12   2 4             2 -14   1               -1 -16 2                 2

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-6$ ${\mathbb {Z} }^{2}$ $r=-5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{4}$ $r=1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

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